Document 1449278

Available online at www.sciencedirect.com
Annals of Physics 323 (2008) 34–48
www.elsevier.com/locate/aop
Efficient wireless non-radiative mid-range
energy transfer
Aristeidis Karalis
a
a,*
, J.D. Joannopoulos b, Marin Soljačić
b
Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology,
77 Massachusetts Avenue, Cambridge, MA 02139, USA
b
Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA
Received 17 April 2007; accepted 17 April 2007
Available online 27 April 2007
Abstract
We investigate whether, and to what extent, the physical phenomenon of long-lifetime resonant
electromagnetic states with localized slowly-evanescent field patterns can be used to transfer energy
efficiently over non-negligible distances, even in the presence of extraneous environmental objects.
Via detailed theoretical and numerical analyses of typical real-world model-situations and realistic
material parameters, we establish that such a non-radiative scheme can lead to ‘‘strong coupling’’
between two medium-range distant such states and thus could indeed be practical for efficient medium-range wireless energy transfer.
2007 Elsevier Inc. All rights reserved.
Keywords: Wireless energy; Wireless power; Strong coupling
1. Introduction
In the early days of electromagnetism, before the electrical-wire grid was deployed, serious interest and effort was devoted (most notably by Nikola Tesla [1]) towards the development of schemes to transport energy over long distances without any carrier medium
(e.g. wirelessly). These efforts appear to have met with little success. Radiative modes of
*
Corresponding author. Fax: +1 617 253 2562.
E-mail address: [email protected] (A. Karalis).
0003-4916/$ - see front matter 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.aop.2007.04.017
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
35
omni-directional antennas (which work very well for information transfer) are not suitable
for such energy transfer, because a vast majority of energy is wasted into free space.
Directed radiation modes, using lasers or highly-directional antennas, can be efficiently
used for energy transfer, even for long distances (transfer distance LTRANS » LDEV, where
LDEV is the characteristic size of the device), but require existence of an uninterruptible
line-of-sight and a complicated tracking system in the case of mobile objects. Rapid
development of autonomous electronics of recent years (e.g. laptops, cell-phones,
house-hold robots, that all typically rely on chemical energy storage) justifies revisiting
investigation of this issue. Today, we face a different challenge than Tesla: since the existing electrical-wire grid carries energy almost everywhere, even a medium-range
(LTRANS few*LDEV) wireless energy transfer would be quite useful for many applications. There are several currently used schemes, which rely on non-radiative modes
(magnetic induction), but they are restricted to very close-range (LTRANS « LDEV) or very
low-power (mW) energy transfers [2–6].
In contrast to all the above schemes, we investigate the feasibility of using long-lived
oscillatory resonant electromagnetic modes, with localized slowly-evanescent field patterns,
for efficient wireless non-radiative mid-range energy transfer. The proposed method is based
on the well known principle of resonant coupling (the fact that two same-frequency resonant objects tend to couple, while interacting weakly with other off-resonant environmental
objects) and, in particular, resonant evanescent coupling (where the coupling mechanism is
mediated through the overlap of the non-radiative near-fields of the two objects). This well
known physics leads trivially to the result that energy can be efficiently coupled between
objects in the extremely near field (e.g. in optical waveguide or cavity couplers and in resonant inductive electric transformers). However, it is far from obvious how this same physics performs at mid-range distances and, to our knowledge, there is no work in the literature
that demonstrates efficient energy transfer for distances a few times larger that the largest
dimension of both objects involved in the transfer. In the present paper, our detailed theoretical and numerical analysis shows that such an efficient mid-range wireless energyexchange can actually be achieved, while suffering only modest transfer and dissipation
of energy into other off-resonant objects, provided the exchange system is carefully
designed to operate in a regime of ‘‘strong coupling’’ compared to all intrinsic loss rates.
The physics of ‘‘strong coupling’’ is also known but in very different areas, such as those
of light-matter interactions [7]. In this favorable operating regime, we quantitatively
address the following questions: up to which distances can such a scheme be efficient and
how sensitive is it to external perturbations? The omnidirectional but stationary (non-lossy)
nature of the near field makes this mechanism suitable for mobile wireless receivers. It could
therefore have a variety of possible applications including for example, placing a source
(connected to the wired electricity network) on the ceiling of a factory room, while devices
(robots, vehicles, computers, or similar) are roaming freely within the room. Other possible
applications include electric-engine buses, RFIDs, and perhaps even nano-robots.
2. Range and rate of coupling
The range and rate of the proposed wireless energy-transfer scheme are the first subjects
of examination, without considering yet energy drainage from the system for use into
work. An appropriate analytical framework for modeling this resonant energy-exchange
is that of the well-known coupled-mode theory (CMT) [8]. In this picture, the field of
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A. Karalis et al. / Annals of Physics 323 (2008) 34–48
the system of two resonant objects 1 and 2 is approximated by F(r, t) a1(t)F1(r) +
a2(t)F2(r), where F1,2(r) are the eigenmodes of 1 and 2 alone, and then the field amplitudes
a1(t) and a2(t) can be shown [8] to satisfy, to lowest order:
da1
dt
da2
dt
¼ iðx1 iC1 Þa1 þ ija2
¼ iðx2 iC2 Þa2 þ ija1
;
ð1Þ
where x1,2 are the individual eigenfrequencies, C1,2 are the resonance widths due to the
objects’ intrinsic (absorption, radiation, etc.) losses, and j is the coupling coefficient.
Eqs. (1) show that at exact resonance (x1 = x2 and C1 = C2), the normal modes of the
combined system are split by 2j; the energy exchange between the two objects takes place
in time p/2k and is nearly perfect, apart for losses, which are minimal when
the coupling
pffiffiffiffiffiffiffiffiffiffi
rate is much faster than all loss rates (j C1,2).1 It is exactly this ratio j= C1 C2 that we
will set as our figure-of-merit for any system under consideration for wireless energy-transfer, along
with the distance over which this ratio can be achieved. The desired optimal repffiffiffiffiffiffiffiffiffiffi
gime j= C1 C2 1 is called ‘‘strong-coupling’’ regime.
Consequently, our energy-transfer application requires resonant modes of high Q = x/
2C for low (slow) intrinsic-loss rates C, and this is why we propose a scheme where the
coupling is implemented using, not the lossy radiative far-field, but the evanescent (nonlossy) stationary near-field. Furthermore, strong (fast) coupling rate j is required over distances larger than the characteristic sizes of the objects, and therefore, since the extent of
the near-field into the air surrounding a finite-sized resonant object is set typically by the
wavelength (and quantified rigorously by the ‘‘radiation caustic’’), this mid-range nonradiative coupling can only be achieved using resonant objects of subwavelength size,
and thus significantly longer evanescent field-tails. This is a regime of operation that
has not been studied extensively, since one usually prefers short tails to minimize interference with nearby devices. As will be seen in examples later on, such subwavelength resonances can often be accompanied with a high radiation-Q, so this will typically be the
appropriate choice for the possibly-mobile resonant device-object d. Note, though, that
the resonant source-object s will in practice often be immobile and with less stringent
restrictions on its allowed geometry and size, which can be therefore chosen large enough
that the near-field extent is not limited by the wavelength (using for example waveguides
with guided modes tuned close to the ‘‘light line’’ in air for slow exponential decay
therein).
The proposed scheme is very general and any type of resonant structure (e.g.
electromagnetic, acoustic, nuclear) satisfying the above requirements can be used for its
implementation. As examples and for definiteness, we choose to work with two
well-known, but quite different, electromagnetic resonant systems: dielectric disks and
capacitively-loaded conducting-wire loops. Even without optimization, and despite their
simplicity, both will be shown to exhibit acceptably good performance.
1
The CMT model is valid exactly for this optimal operational regime of well-defined resonances. Its range of
applicability does not include very-close-distance coupling, since there the necessary condition j « x1,2 does not
hold, neither large-distance far-field coupling, since it fails to predict far-field interference effects and accurate
radiation patterns; rather CMT is exactly suitable for the medium-distance near-field coupling of our interest.
Thus the use of this model is justified and the parameters j, C1,2 are well defined.
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
37
2.1. Dielectric disks
Consider a 2D dielectric disk object of radius r and relative permittivity e surrounded by
air that supports high-Q ‘‘whispering-gallery’’ resonant modes (Fig. 1). The loss mechanisms for the energy stored inside such a resonant system are radiation into free space
and absorption inside the disk material. High-Qrad and long-tailed subwavelength resonances can be achieved, only when the dielectric permittivity e is large and the azimuthal
field variations are slow (namely of small principal number m). Material absorption is
related to the material loss tangent: Qabs Re{e}/Im{e}. Mode-solving calculations for
this type of disk resonances were performed using two independent methods: numerically,
2D finite-difference frequency-domain (FDFD) simulations (which solve Maxwell’s Equations in frequency domain exactly apart for spatial discretization) were conducted with a
Fig. 1. Main plot. A 2D high-e disk of radius r (shown in yellow) surrounded by air, along with the electric field
(with polarization pointing out of the page) of its resonant whispering-gallery mode superimposed (shown in red/
white/blue in regions of positive/zero/negative field respectively). Side plot. Radial plot of the electric field of the
mode shown in the main plot (basically a cross-section of the main plot). Note that in air (radius/r > 1) the field
follows a Hankel-function form, with an initial exponential-like regime (with long tails compared to the small disk
size), followed by the oscillatory/radiation regime (whose presence means that energy is slowly leaking out of the
disk). Table. Numerical FDFD (and in parentheses analytical SV) results for the wavelength and absorption,
radiation and total loss rates, for two different cases of subwavelength-disk resonant modes. Note that diskmaterial loss-tangent Im{e}/Re{e}=104 was used. (The specific parameters of the plot are highlighted with bold in
the table.) Finally, note that for the 3D case the computational complexity would be immensely increased, while
the physics would not be significantly different. For example, a spherical object of e = 147.7 has a whispering
gallery mode with m = 2, Qrad = 13,962, and k/r = 17. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this paper.)
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A. Karalis et al. / Annals of Physics 323 (2008) 34–48
resolution of 30 pts/r; analytically, standard separation of variables (SV) in polar coordinates was used. The results for two TE-polarized dielectric-disk subwavelength modes of
k/r P 10 are presented in Fig. 1. The two methods have excellent agreement and imply
that for a properly designed resonant low-loss-dielectric object values of Qrad P 2000
and Qabs 10,000 should be achievable.
Note that the required values of e, shown in Fig. 1, might at first seem unrealistically
large. However, not only are there in the microwave regime (appropriate for meter-range
coupling applications) many materials that have both reasonably high enough dielectric
constants and low losses (e.g. Titania: e 96, Im{e}/e 103; Barium tetratitanate:
e 37, Im{e}/e 104; Lithium tantalite: e 40, Im{e}/e 104; etc.) [9,10], but also e
could signify instead the effective index of other known subwavelength (k/r 1) surface-wave systems, such as surface-plasmon modes on surfaces of metal-like (negative-e)
materials [11] or metallo-dielectric photonic crystals [12].
To calculate now the achievable rate of energy transfer between two disks 1 and 2, we
place them at distance D between their centers (Fig. 2). Numerically, the FDFD mode-solver simulations give j through the frequency splitting (=2j) of the normal modes of the
combined system, which are even and odd superpositions of the initial single-disk modes;
analytically, using the expressions
for the separation-of-variables
eigenfields E1,2(r) CMT
R
R
2
gives j through j ¼ x1 =2 d3 re2 ðrÞE2 ðrÞE1 ðrÞ= d3 reðrÞjE1 ðrÞj , where ej(r) and e(r) are
the dielectric functions that describe only the disk j (minus the constant eo background)
and the whole space respectively. Then, for medium distances D/r = 10–3 and for nonradiative coupling such that D < 2rC, where rC = mk/2p is the radius of the radiation caustic, the two methods agree very well, and we finally find (Fig. 2) coupling-to-loss ratios in
the range j/C 1–50. Although the achieved figure-of-merit values do not fall in the ideal
‘‘strong-coupling’’ operating regime j/C 1, they are still large enough to be useful for
applications, as we will see later on.
2.2. Capacitively-loaded conducting-wire loops
Consider a loop of radius r of conducting wire with circular cross-section of radius a
connected to a pair of conducting parallel plates of area A spaced by distance d via a
dielectric of relative permittivity e and everything surrounded by air (Fig. 3). The wire
has inductance L, the plates have capacitance C and then the system has a resonant mode,
where the nature of the resonance lies in the periodic exchange of energy from the electric
field inside the capacitor, due to the voltage across it, to the magnetic field in free space,
due to the current in the wire. Losses in this resonant system consist of ohmic loss Rabs
inside the wire and radiative loss Rrad into free space. Mode-solving calculations for this
type of RLC-circuit resonances were performed using again two independent methods:
numerically, 3D finite-element frequency-domain (FEFD) simulations (which solve Maxwell’s Equations in frequency domain exactly apart for spatial discretization) were conducted [13], in which
theffi boundaries of the conductor were modeled using a complex
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
impedance gc ¼ lc x=2r boundary condition, valid
as long as gc/go 1 [14] (<105
pffiffiffiffiffiffiffiffiffiffiffi
for copper in the microwave), where lo, eo and go ¼ lo =eo are the magnetic permeability, electric permittivity and impedance of free space and r is the conductivity of the conductor; analytically, the formulas L = lor [ln(8r/a) 2] [15] and C = eoeA/d, and, in the
desired subwavelength-loop (r « k) limit, the quasi-static formulas Rabs gcr/a (which
takes skin-depth effects into account) and Rrad p/6go(r/k)4 [15] were used to determine
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
39
Fig. 2. Plot. System of two same 2D high-e disks of radius r (yellow) for medium-distance D coupling between
them, along with the electric field of the normal mode, which is an even superposition of the single-disk modes of
Fig. 1, superimposed (red/white/blue). Note that there is also a normal mode, which is an odd superposition of
the single-disk modes of Fig. 1 (not shown). Table. Numerical FDFD (and in parentheses analytical CMT) results
for the average of the wavelength and loss rates of the two normal modes (individual values not shown), and also
the coupling rate and ‘‘strong/weak-coupling’’ figure-of-merit as a function of the coupling distance D, for the
two cases of disk modes presented in Fig. 1. Only distances for non-radiative (D < 2rC) coupling are considered.
Note that the average Crad (and thus total C) shown are slightly different from the single-disk value of Fig. 1, due
to far-field interference effects present for the two normal modes, for which CMT cannot make predictions and
this is why analytical results for Crad are not shown but the single-disk value is used. (The specific parameters of
the plot are highlighted with bold in the table.) (For interpretation of the references to color in this figure legend,
the reader is referred to the web version of this paper.)
pffiffiffiffiffiffiffi
the resonant frequency x ¼ 1= LC and its quality factors Qabs = xL/Rabs and Qrad = xL/
Rrad. By tuning the capacitance and thus the resonant frequency, the total Q becomes
highest for some optimal frequency determined by the loop parameters: at low frequencies
it is dominated by ohmic loss and at high frequencies by radiation. The results for two subwavelength modes of k/r P 70 (namely highly suitable for near-field coupling and really in
the quasi-static limit) at this optimal frequency are presented in Fig. 3. The two methods
are again in very good agreement and show that expected quality factors in the microwave
are Qabs P 1000 and Qrad P 10,000.
For the rate of energy transfer between two loops 1 and 2 at distance D between their
centers (Fig. 4): numerically, the FEFD mode-solver simulations give j again through the
frequency splitting (=2j) of the normal modes of the combined system; analytically, j is
40
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
Fig. 3. Plot. A wire loop of radius r connected to a pair of d-spaced parallel plates (shown in yellow) surrounded
by air, along with a slice of the magnetic field (component parallel to the axis of the circular loop) of their
resonant mode superimposed (shown in red/white/blue in regions of positive/zero/negative field respectively).
Table. Numerical FEFD (and in parentheses analytical) results for the wavelength and absorption, radiation and
total loss rates, for two different cases of subwavelength-loop resonant modes. Note that for conducting material
copper (r = 5.998 · 107 S/m) was used. (The specific parameters of the plot are highlighted with bold in the table.)
(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this
paper.)
pffiffiffiffiffiffiffiffiffi
given by j ¼ xM=2 L1 L2 , where M is the mutual inductance of the two loops, which, in
the quasi-static limit r « D « k and for the relative orientation shown in Fig. 4, is M p/
pffiffiffiffiffiffiffiffi 3
2lo(r1r2)2/D3 [14], which means that x=2j ðD= r1 r2 Þ . Then, and for medium distances
D/r = 10–3, the two methods agree well, and we finally find (Fig. 4) coupling-to-loss
ratios, which peak at a frequency between those where the single-loop Q1,2 peak and
are in the range j/C 0.5–50.
It is important to appreciate the difference between such a resonant-coupling inductive
scheme and the well-known non-resonant inductive scheme for energy transfer. Using
CMT it is easy to show that, keeping the geometry and the energy stored at the source
fixed, the resonant inductive mechanism allows for Q2 (106) times more power delivered for work at the device than the traditional non-resonant mechanism. This is why only
close-range contact-less medium-power (W) transfer is possible with the latter [2,3],
while with resonance either close-range but large-power (kW) transfer is allowed [4,5]
or, as currently proposed, if one also ensures operation in the strongly-coupled regime,
medium-range and medium-power transfer is possible. Capacitively-loaded conductive
loops are actually being widely used also as resonant antennas (for example in cell
phones), but those operate in the far-field regime with D/r » 1,r/k 1, and the radiation
Q’s are intentionally designed to be small to make the antenna efficient, so they are not
appropriate for energy transfer.
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
41
Fig. 4. Plot. System of two same wire loops connected to parallel plates (yellow) for medium-distance D coupling
between them, along with a slice of the magnetic field of the even normal mode superimposed (red/white/blue).
Note that there is also an odd normal mode (not shown). Table. Numerical FEFD (and in parentheses analytical)
results for the average wavelength and loss rates of the two normal modes (individual values not shown), and also
the coupling rate and ‘‘strong/weak-coupling’’ figure-of-merit as a function of the coupling distance D, for the
two cases of modes presented in Fig. 3. Note that the average Crad shown are again slightly different from the
single-loop value of Fig. 3, due to far-field interference effects present for the two normal modes, which again the
analytical model cannot predict and thus analytical results for Crad are not shown but the single-loop value is
used. (The specific parameters of the plot are highlighted with bold in the table.) (For interpretation of the
references to color in this figure legend, the reader is referred to the web version of this paper.)
3. Influence of extraneous objects
Clearly, the success of the proposed resonance-based wireless energy-transfer scheme
depends strongly on the robustness of the objects’ resonances. Therefore, their sensitivity
to the near presence of random non-resonant extraneous objects is another aspect of the
proposed scheme that requires analysis. The appropriate analytical model now is that of
perturbation theory (PT) [8], which suggests that in the presence of an extraneous object
e the field amplitude a1(t) inside the resonant object 1 satisfies, to first order:
da1
¼ iðx1 iC1 Þa1 þ iðj11e þ iC1e Þa1
dt
ð2Þ
42
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
where again x1 is the frequency and C1 the intrinsic (absorption, radiation, etc.) loss rate,
while j11e is the frequency shift induced onto 1 due to the presence of e and C1e is the
extrinsic due to e (absorption inside e, scattering from e, etc.) loss rate2. The frequency
shift is a problem that can be ‘‘fixed’’ rather easily by applying to every device a feedback
mechanism that corrects its frequency (e.g. through small changes in geometry) and
matches it to that of the source. However, the extrinsic loss can be detrimental to the functionality of the energy-transfer scheme, because it cannot be remedied,
so the
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi total loss
rate C1[e] = C1 + C1e and the corresponding figure-of-merit j½e = C1½e C2½e , where j[e]
the perturbed coupling rate, must be quantified.
3.1. Dielectric disks
In the first example of resonant objects that we have considered, namely dielectric disks,
small, low-index, low-material-loss or far-away stray objects will induce small scattering
and absorption. In such cases of small perturbations these extrinsic loss mechanisms
can be quantified
using respectively the analytical
R 3
R 3 first-order PT 2 formulas
2
2
abs
Crad
/
x
d
rjRefe
ðrÞgj
jE
ðrÞj
=U
and
C
¼
x
=4
d rImfee ðrÞgjE1 ðrÞj =U , where
1
e
1
1
1e
1e
R
2
U ¼ 1=2 d3 reðrÞjE1 ðrÞj is the total resonant electromagnetic energy of the unperturbed
mode. As one can see, both of these losses depend on the square of the resonant electric
field tails E1 at the site of the extraneous object. In contrast, the
R coupling rate from object
1 to another resonant object 2 is, as stated earlier, j ¼ x1 =4 d3 re2 ðrÞE2 ðrÞE1 ðrÞ=U and
depends linearly on the field tails E1 of 1 inside 2. This difference in scaling gives us confidence that, for exponentially small field tails, coupling to other resonant objects should
be much faster than all extrinsic loss rates (j C1e), at least for small perturbations, and
thus the energy-transfer scheme is expected to be sturdy for this class of resonant dielectric
disks.
However, we also want to examine certain possible situations where extraneous objects
cause perturbations too strong to analyze using the above first-order PT approach. For
example, we place a dielectric disk c close to another off-resonance object of large
Re{e}, Im{e} and of same size but different shape (such as a human being h), as shown
in Fig. 5a, and a roughened surface of large extent but of small Re{e}, Im{e} (such as a
wall w), as shown in Fig. 5b. For distances Dh/w/r = 10–3 between the disk-center and
the ‘‘human’’-center/‘‘wall’’, the numerical FDFD simulation results presented in Fig. 5
rad
rad
abs
suggest that Qrad
c½h ; Qc½w P 1000 (instead of the initial Qc P 2000Þ, Qc 10; 000 (natu5
2
5
4
abs
rally unchanged), Qabs
ch 10 –10 , and Qcw 10 –10 , namely the disk resonance seems
to be fairly robust, since it is not detrimentally disturbed by the presence of extraneous
objects, with the exception of the very close proximity of high-loss objects [16].
To examine the influence of large perturbations on an entire energy-transfer system we
consider two resonant disks in the close presence of both a ‘‘human’’ and a ‘‘wall’’. The
numerical FDFD simulations show that the system performance deteriorates from j/
Cc 1–50 (Fig. 2) to j[hw]/Cc[hw] 0.5–10 (Fig. 6), i.e. only by acceptably small amounts.
2
The first-order PT model is valid only for small perturbations. Nevertheless, the parameters j11e, C1e are
well defined, even outside that regime, if a1 is taken to be the amplitude of the exact perturbed mode.
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
43
Fig. 5. Plots. A disk (yellow) in the proximity at distance Dh/w of an extraneous object (yellow): (a) a high
e = 49 + 16i (which is large but actually appropriate for human muscles in the GHz regime [16]) square object of
same size (area) with the disk, and (b) a large roughened surface of e = 2.5 + 0.05i (appropriate for ordinary
materials such as concrete, glass, plastic, wood [16]), along with the electric field of the disk’s perturbed resonant
mode superimposed (red/white/blue). Tables. Numerical FDFD results for the parameters of the disk’s perturbed
resonance, including absorption rate inside the extraneous object and total (including scattering from the
extraneous object) radiation-loss rate, for the two cases of disk modes presented in previous figures. Note that
again disk-material loss-tangent Im{e}/Re{e}=104 was used, and that Qrad
c½h=w is again different (decreased or even
increased) from the single-disk Qrad
of Fig. 1, due to (respectively constructive or destructive) interference effects
c
this time between the radiated and strongly scattered far-fields. (The specific parameters of the plots are highlighted
with bold in the tables.) (For interpretation of the references to color in this figure legend, the reader is referred to
the web version of this paper.)
3.2. Capacitively-loaded conducting-wire loops
In the second example of resonant objects that we have considered, the conducting-wire
loops, the influence of extraneous objects on the resonances is nearly absent. The reason is
that, in the quasi-static regime of operation (r « k) that we are considering, the near field in
the air region surrounding the loop is predominantly magnetic (since the electric field is
localized inside the capacitor), therefore extraneous non-metallic objects e that could interact with this field and act as a perturbation to the resonance are those having significant
magnetic properties (magnetic permeability Re{l} > 1 or magnetic loss Im{l} > 0). Since
almost all every-day materials are non-magnetic, they respond to magnetic fields in the
same way as free space, and thus will not disturb the resonance of a conducting-wire loop.
To get only a rough estimate of this disturbance, we use the PT formula, stated earlier,
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A. Karalis et al. / Annals of Physics 323 (2008) 34–48
Fig. 6. Plot. System of two same disks (yellow) for medium-distance D coupling between them in the proximity at
equal distance D of two extraneous objects (yellow): both a high e = 49 + 16i square object of same size (area)
with the disks and a large roughened surface of e = 2.5 + 0.05i, along with the electric field of the system’s
perturbed even normal mode superimposed (red/white/blue). Table. Numerical FDFD results for the average
wavelength and loss rates of the system’s perturbed two normal modes (individual values not shown), and also the
perturbed coupling rate and ‘‘strong/weak-coupling’’ figure-of-merit as a function of the distance D, for the two
cases of disk modes presented in previous Figures. Only distances for non-radiative (D < 2rC) coupling are
considered. Note once more that the average Crad takes into account interference effects between all radiated and
scattered far-fields. (The specific parameters of the plot are highlighted with bold in the table.) (For interpretation of
the references to color in this figure legend, the reader is referred to the web version of this paper.)
R 3
2
Cabs
d r Imfee ðrÞgjE1 ðrÞj =U with the numerical results for the field of an exam1e ¼ x1 =4 ple like the one shown in the plot of Fig. 4 and with a rectangular object of dimensions
30 cm · 30 cm · 1.5 m and permittivity e = 49 + 16i (human muscles) residing between
the loops and almost standing on top of one capacitor (3 cm away from it) and find
5
5
abs
Qabs
ch 10 and for 10 cm away Qch 5 10 . Thus, for ordinary distances (1 m)
and placements (not immediately on top of the capacitor) or for most ordinary extraneous
objects e of much smaller loss-tangent, we conclude that it is indeed fair to say that
Qabs
ce ! 1 and that j[e]/C[e] j/C 0.5–50. The only perturbation that is expected to
affect these resonances is a close proximity of large metallic structures.
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
45
An extremely important implication of this fact relates to safety considerations for
human beings. Humans are also non-magnetic and can sustain strong magnetic fields without undergoing any risk. A typical example, where magnetic fields B 1 T are safely used
on humans, is the magnetic resonance imaging (MRI) technique for medical testing. In
contrast, the magnetic near-field required by our scheme in order to provide a few Watts
of power to devices is only B 104 T, which is actually comparable to the magnitude of
the Earth’s magnetic field. Since, as explained above, a strong electric near-field is also not
present and the radiation produced from this non-radiative scheme is minimal, it is reasonable to expect that our proposed energy-transfer method should be safe for living
organisms.
In comparison of the two classes of resonant systems under examination, the strong
immunity to extraneous objects and the absence of risks for humans probably makes
the conducting-wire loops the preferred choice for many real-world applications; on the
other hand, systems of disks (or spheres) of high (effective) refractive index have the
advantage that they are also applicable to much smaller length-scales (for example in
the optical regime dielectrics prevail, since conductive materials are highly lossy).
4. Efficiency of energy-transfer scheme
Consider again the combined system of a resonant source s and device d in the presence
of a set of extraneous objects e, and let us now study the efficiency of this resonance-based
energy-transfer scheme, when energy is being drained from the device at rate Cwork for use
into operational work. The coupled-mode-theory equation for the device field-amplitude is
dad
¼ iðx iCd½e Þad þ ij½e as Cwork ad ;
dt
ð3Þ
abs
rad
abs
abs
where Cd½e ¼ Crad
d½e þ Cd½e ¼ Cd½e þ ðCd þ Cde Þ is the net perturbed-device loss rate, and
similarly we define Cs[e] for the perturbed-source. Different temporal schemes can be used
to extract power from the device (e.g. steady-state continuous-wave drainage, instantaneous drainage at periodic times and so on) and their efficiencies exhibit different dependence on the combined system parameters. Here, we assume steady state, such that the
field amplitude inside the source is maintained constant, namely as(t) = Aseixt, so then
the field amplitude inside the device is ad(t) = Adeixt with Ad =As ¼ ij½e =ðCd½e þ Cwork Þ.
The various time-averaged powers of interest are then: the useful extracted power is
2
Pwork = 2CworkjAdj2, the radiated (including scattered) power is P rad ¼ 2Crad
s½e jAs j þ
2
2
abs
2Crad
d½e jAd j , the power absorbed at the source/device is P s=d ¼ 2Cs=d jAs=d j , and at the extra2
2
abs
abs
neous objects P e ¼ 2Cse jAs j þ 2Cde jAd j . From energy conservation, the total time-averaged power entering the system is Ptotal = Pwork + Prad + Ps + Pd + Pe. Note that the
reactive powers, which are usually present in a system and circulate stored energy around
it, cancel at resonance (which can be proven for example in electromagnetism from Poynting’s Theorem [14]) and do not influence the power-balance calculations. The working efficiency is then:
gwork P work
¼
P total
C
1
d½e
1 þ fom1 2
1 þ Cwork
½e
2 ;
1 þ CCwork
d½e
ð4Þ
46
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
where fom½e ¼ j½e = Cs½e Cd½e is the distance-dependent figure-of-merit of the perturbed
resonant energy-exchange system. Depending on the targeted application, reasonable
choices for the work-drainage rate q
are:
Cwork/Cd[e] = 1 to minimize the required energy
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 þ fom2½e > 1 to maximize the efficiency for some
stored in the source, Cwork =Cd½e ¼
particular value of fom[e] or Cwork/Cd[e] 1 to minimize the required energy stored in
the device. For any of these choices, gwork is a function of the fom[e] parameter only. gwork
is shown for its optimal choice in Fig. 7 with a solid black line, and is gwork > 17% for
fom[e] > 1, namely large enough for practical applications. The loss conversion ratios depend also on the other system parameters, and the most disturbing ones (radiation and
absorption in stray objects) are plotted in Fig. 7 for the two example systems of dielectric
disks and conducting loops with values for their parameters within the ranges determined
earlier.
To get a numerical estimate for a system performance, take, for example, coupling distance D/r = 5, a ‘‘human’’ extraneous object at distance Dh/r = 10, and that Pwork = 10 W
must be delivered to the load. Then, for dielectric disks we have (based on Fig. 5)
3
4
4
rad
abs
abs
abs
Qrad
¼ Qabs
s½h ¼ Qd½h 10 , Qs
d 10 , Qsh ¼ Qdh 5 10 and (based on Figs. 2 and
6) fom[h] 3, so from Fig. 7 we find efficiency gwork = 52% and that Prad 8.3 W will
1
0.9
radiation loss
for dielectric disks
Q decreases
conversion ratios (efficiencies)
0.8
e
efficiency
ηwork
0.7
0.6
0.5
0.4
0.3
0.2
radiation loss
for conducting loops
absorption loss in e
for dielectric disks
Qe decreases
0.1
0
.1
.2
.3
.5
1
2
3
5
10
20
30
50
100
1/2
fom[e] = κ[e]/(Γs[e]Γd[e])
Fig. 7. Black line. Efficiency of converting the supplied power into useful work (gwork) as a function of the
perturbed coupling-to-loss figure-of-merit, optimized with respect to the power-extracting rate Cwork (related to
the load impedance), for all values of the various quality factors that are present in the system. Blue and red lines.
Ratios of power conversion respectively into radiation (including scattering from nearby extraneous objects) and
3
rad
dissipation inside an extraneous object as a function of the figure-of-merit for dielectric disks of Qrad
s½e ¼ Qd½e 10
4
4
4
5
abs
abs
abs
and Qabs
¼
Q
10
,
and
for
three
values
of
Q
¼
Q
¼
10
;
5
10
;
10
.
Green
line.
Ratio
of
power
s
d
se
de
4
3
rad
abs
conversion into radiation for conducting-wire loops of Qrad
¼ Qabs
s½e ¼ Qd½e 10 and Qs
d 10 , and assuming
abs
abs
Qse ¼ Qde ! 1. (For interpretation of the references to color in this figure legend, the reader is referred to the
web version of this paper.)
A. Karalis et al. / Annals of Physics 323 (2008) 34–48
47
be radiated to free space, Ps 0.5 W will be dissipated inside the source, Pd 0.3 W inside
the device, and Ph 0.2 W inside the human. On the other hand, for conducting loops we
4
3
rad
abs
abs
abs
have (based on Figs. 3 and 4) Qrad
¼ Qabs
s½h ¼ Qd½h 10 , Qs
d 10 , Qsh ¼ Qdh ! 1 and
fom[h] 4, so we find gwork = 61%, Prad 0.6 W, Ps 3.6 W, Pd 2.2 W, and most
importantly Ph fi 0.
5. Conclusion
In conclusion, we present a scheme based on ‘‘strongly-coupled’’ resonances for midrange wireless non-radiative energy transfer. Although our consideration has been for a
static geometry (namely j and Ce were independent of time), all the results can be applied
directly for the dynamic geometries of mobile objects, since the energy-transfer time j1
(1–100 ls for microwave applications) is much shorter than any timescale associated
with motions of macroscopic objects. Analyses of very simple implementation geometries
provide encouraging performance characteristics and further improvement is expected
with serious design optimization. Thus the proposed mechanism is promising for many
modern applications. For example, in the macroscopic world, this scheme could potentially be used to deliver power to robots and/or computers in a factory room, or electric
buses on a highway (source-cavity would in this case be a ‘‘pipe’’ running above the highway). In the microscopic world, where much smaller wavelengths would be used and smaller powers are needed, one could use it to implement optical inter-connects for CMOS
electronics, or to transfer energy to autonomous nano-objects (e.g. MEMS or nanorobots) without worrying much about the relative alignment between the sources and
the devices.
As a venue of future scientific research, enhanced performance should be pursued for
electromagnetic systems either by exploring different materials, such as plasmonic or metallodielectric structures of large effective refractive index, or by fine-tuning the system
design, for example by exploiting the earlier mentioned interference effects between the
radiation fields of the coupled objects. Furthermore, the range of applicability could be
extended to acoustic systems, where the source and device are connected via a common
condensed-matter object.
Acknowledgments
We thank Prof. John Pendry and L.J. Radziemski for suggesting magnetic and acoustic
resonances respectively, and Prof. Steven G. Johnson, Prof. Peter Fisher, André Kurs and
Miloš Popović for useful discussions. This work was supported in part by the Materials
Research Science and Engineering Center program of the National Science Foundation
under Grant No. DMR 02-13282, by the US Department of Energy under Grant No.
DE-FG02-99ER45778, and by the Army Research Office through the Institute for Soldier
Nanotechnologies under Contract No. DAAD-19-02-D0002.
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